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How can I find the Galois group of $(x^2-2)(x^3+x+1)$?

I read the answer to this question for techniques but I didn't find anything about the product of two polynomials.

For the first factor the group is clearly $Z_{2}$, but I can't see it for the second one (how does it help that it has exactly one real root?). Is it generally true that the product of two irreducibles has a Galois group that is the direct sum of the factors?

sharding4
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Nesa
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  • Think about the composite of the splitting fields of each of the polynomials. Check this post https://math.stackexchange.com/q/35429/254075 – sharding4 May 23 '17 at 23:31

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